How should we pool effect sizes in meta-analysis?
2024-07-30
To get our meta-analysis result, we need to take a weighted average of effect sizes…
But how should we weigh the evidence of each study?
Study weight \(w_{k}\) is inversely related to the variance (\(SE^2_{k}\))
\[ w_{k}=\frac{1}{(SE)^{2}_{k}} \]
The pooled estimate \(\widehat{\theta}\) is the weighted average:
\[ \widehat{\theta} = \frac{\Sigma_{k=1}^{K} \widehat{\theta}_{k} w_{k}}{\Sigma_{k=1}^{K} w_{k}} \]
To compute the standard error on our averaged result \(\widehat\theta\)
\[ SE_{k} = \frac{1}{\Sigma(w)} \]
A study’s effect size (\(\widehat{\theta}_{k}\)) is an estimate of the true study effect size (\(\theta_{k}\)) plus sampling error (\(\epsilon_{k}\)).
\[ \widehat{\theta}_{k}=\theta_{k} +\epsilon_{k} \]
The true effect size of study \(k\) (\(\theta_{k}\)) is drawn from a distribution with mean \(\mu\) and has the new error term \(\zeta_{k}\).
\[ {\theta}_{k}=\mu+\zeta_{k} \]
\[ w^{*}_{k}=\frac{1}{SE^{2}_{k} + \tau^2} \]
Then compute the weighted average in the usual fashion.
Only choose the FEM if you have clear reason to (rare)
For most scenarios, use the REM.
My slides are online here: https://jmoggridge.github.io/fixed-and-random-effects